होम
Journal of Philosophical Logic The Logic of Pragmatic Truth
The Logic of Pragmatic Truth
Newton C. A. da Costa, Otávio Bueno, Steven Frenchयह पुस्तक आपको कितनी अच्छी लगी?
फ़ाइल की गुणवत्ता क्या है?
पुस्तक की गुणवत्ता का मूल्यांकन करने के लिए यह पुस्तक डाउनलोड करें
डाउनलोड की गई फ़ाइलों की गुणवत्ता क्या है?
खंड:
27
भाषा:
english
पृष्ठ:
18
DOI:
10.1023/a:1004304228785
Date:
December, 1998
फ़ाइल:
PDF, 112 KB
आपके टैग:
फ़ाइल 15 मिनट के भीतर आपके ईमेल पते पर भेजी जाएगी.
फ़ाइल 15 मिनट के भीतर आपकी Kindle पर डिलीवर हो जाएगी.
टिप्पणी: आप जो भी पुस्तक अपने Kindle पर भेजना चाहें इसे सत्यापित करना होगा. Amazon Kindle Support से सत्यापन ईमेल के लिए अपना मेलबॉक्स देखें.
टिप्पणी: आप जो भी पुस्तक अपने Kindle पर भेजना चाहें इसे सत्यापित करना होगा. Amazon Kindle Support से सत्यापन ईमेल के लिए अपना मेलबॉक्स देखें.
Conversion to is in progress
Conversion to is failed
0 comments
आप पुस्तक समीक्षा लिख सकते हैं और अपना अनुभव साझा कर सकते हैं. पढ़ूी हुई पुस्तकों के बारे में आपकी राय जानने में अन्य पाठकों को दिलचस्पी होगी. भले ही आपको किताब पसंद हो या न हो, अगर आप इसके बारे में ईमानदारी से और विस्तार से बताएँगे, तो लोग अपने लिए नई रुचिकर पुस्तकें खोज पाएँगे.
1


2


NEWTON C. A. DA COSTA, OTÁVIO BUENO and STEVEN FRENCH THE LOGIC OF PRAGMATIC TRUTH1 In memory of Rolando Chuaqui ABSTRACT. The mathematical concept of pragmatic truth, first introduced in Mikenberg, da Costa and Chuaqui (1986), has received in the last few years several applications in logic and the philosophy of science. In this paper, we study the logic of pragmatic truth, and show that there are important connections between this logic, modal logic and, in particular, Jaskowski’s discussive logic. In order to do so, two systems are put forward so that the notions of pragmatic validity and pragmatic truth can be accommodated. One of the main results of this paper is that the logic of pragmatic truth is paraconsistent. The philosophical import of this result, which justifies the application of pragmatic truth to inconsistent settings, is also discussed. KEY WORDS: Jaskowski’s logic, modal logic, paraconsistent logic, partial structures, pragmatic truth, quasitruth I NTRODUCTION In Mikenberg, da Costa and Chuaqui (1986), the mathematical concept of pragmatic truth was introduced, an infinitary logical system was presented to treat this concept, and some applications of it made in logic and in algebra. An application of this concept in the foundations of the theory of probability is studied in da Costa (1986), and some extensions of it to inductive logic and to the philosophy of science are considered, respectively, in da Costa and French (1989), and da Costa and French (1990). This framework was also employed in order to examine some issues involved in the theory of acceptance (da Costa and French (1993a)), as well as in the modelling of ‘natural reasoning’ (da Costa and French (1993b)). The wide range of applications of the notion of pragmatic truth motivates an investigation of the logic of pragmatic truth. In this paper, we will show that there are important connections between this logic and modal logic, in particular between the logic of pragmatic truth and Jaskowski’s discussive logic (see Jaskows; ki (1969) and da Costa (1975)).2 We will study these connections in the context of da Costa’s formulation of pragmatic structures (cf. da Costa (1986)), which we shall present in Section 1. As we shall see in Section 2, such structures can be considered as worlds Journal of Philosophical Logic 27: 603–620, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands. VTEX(VR) PIPS No.: 154813 (logikap:humnfam) v.1.15 LOGI414.tex; 13/11/1998; 12:00; p.1 604 NEWTON C. A. DA COSTA ET AL. of a Kripke structure, and in this setting the necessity operator in modal logic corresponds to the notion of pragmatic validity, and the possibility operator to the notion of pragmatic truth. Two systems are then put forward in order to study these notions (respectively, in Sections 2 and 3). One of the main results of the present paper, proved in Section 3, is that the logic of pragmatic truth is paraconsistent. The philosophical import of this result, which justifies the application of pragmatic truth to inconsistent settings, is then discussed in Section 4. A remark on our terminology is important here. We call the kind of truth defined in this paper pragmatic truth, owing to its connections with the pragmatic conception of truth, as developed by philosophers like James, Dewey and particularly Peirce (cf. Mikenberg etal. (1986), da Costa (1986), and da Costa and French (forthcoming), Chapter 1). However, our piece is not exegetical. The sole point we would like to emphasize is that our definition was heuristically inspired by some passages of pragmatic thinkers, such as Peirce, when he wrote that, “consider what effects, that might conceivably have practical bearings, we conceive the object of our conceptions to have. Then, our conception of these effects is the whole of our conception of the object” (Peirce (1965), p. 31). In our opinion, the definition of pragmatic truth studied in this paper captures, at least in part, the common concept of a theory saving the appearances, usually by means of partially fictitious constructions (see Vaihinger (1952), and Bueno (1997)). Maybe it would be better to call our kind of truth quasitruth, instead of pragmatic truth. 1. S IMPLE PRAGMATIC STRUCTURES In order to motivate the study of the settheoretical structures which we shall call here simple pragmatic structures, we begin with some informal considerations. Let us suppose that we are interested in studying a certain domain of knowledge 1 in the field of empirical sciences, for instance the physics of particles. We are, then, concerned with certain real objects (in the physics of particles, with some configurations in a Wilson chamber, some spectral lines etc.), whose set we denote by A1 . Among the objects of A1 , there are some relations that interest us and that we model as partial relations Ri , i ∈ I , every relation having a fixed arity. The relations Ri , i ∈ I , are partial relations, that is, each Ri , supposed of arity ri , is not necessarily defined for all ri tuples of elements of A1 , i ∈ I .3 The reason for using partial relations is that they are supposed to express what we do know, or what we accept as true, about the actual relations linking LOGI414.tex; 13/11/1998; 12:00; p.2 THE LOGIC OF PRAGMATIC TRUTH 605 the elements of A1 . Then, the partial structure hA1 , Ri ii∈I encompasses, so to say, what we know or accept as true about the actual structure of 1 (for further formal details, see Mikenberg et al. (1986) and da Costa (1986)). However, in order to systematize our knowledge of 1, it is convenient to introduce in our structure hA1 , Ri ii∈I some ideal objects (in the physics of particles, quarks for example). The set of these new objects will be denoted by A2 . It is understood that A1 ∩A2 = ∅, and we put A = A1 ∪A2 . This way, the modelling of 1 involves new partial relations Rj , j ∈ J , some of which extend the relations Ri , i ∈ I . Furthermore, there are some sentences (closed formulas) of the language L, in which we talk about the structure hA, Rk ik∈I ∪J (I ∩ J = ∅) that we accept as true or that are true (in the sense of the correspondence theory of truth). This occurs, for instance, with sentences expressing true decidable propositions (a proposition whose truth or falsehood can be decided), and with some general sentences which express laws or theories already accepted as true. Let us denote the set of such sentences, dubbed primary, by P . Then, taking into account the above informal discussion, we are led to suggest that a simple pragmatic structure be regarded as a settheoretic structure of the form A = hA1 , A2 , Ri , Rj , P ii∈I,j ∈J where the nonempty sets A1 and A2 , the partial relations Ri and Rj , i ∈ I and j ∈ J , and the set of sentences P satisfy the preceding conditions. More formally, such a structure can be defined as follows: DEFINITION 1. A simple pragmatic structure (sps) is a structure A = hA, Rk , P ik∈K , where A is a nonempty set, Rk is a partial relation defined on A for every k ∈ K, and P is a set of sentences of a language L of the same similarity type as that of A and which is interpreted in A. The set A is called the universe of A. (For some k, Rk may be empty; P may also be empty.) From now on, a sps will always be a structure according to Definition 1. (In order to simplify the exposition, partially defined functions are not included in a sps.) Let L be a firstorder language with equality, but without function symbols. The symbols of L are, then, logical symbols (connectives, individual variables, the quantifiers, and the equality symbol), auxiliary symbols LOGI414.tex; 13/11/1998; 12:00; p.3 606 NEWTON C. A. DA COSTA ET AL. (parentheses), a collection of individual constants, and a collection of predicate symbols. To interpret L in a sps A = hA, Rk , P ik∈K is to associate with each individual constant of L an element of the universe of A, and with each predicate symbol of L of arity n a relation Rk , k ∈ K, of the same arity. It is supposed that every predicate of the family (Rk )k∈K is associated with a predicate symbol. DEFINITION 2. Let L and A = hA, Rk , P ik∈K be respectively a language and a sps, such that L is interpreted in A. Let S be a total structure, that is a usual structure (whose relations of arity n are defined for all ntuples of elements of its universe), and we suppose that L is also interpreted in S. Then, S is said to be Anormal if the following properties are verified: (1) The universe of S is A. (2) The (total) relations of S extend the corresponding partial relations of A. (3) If c is an individual constant of L, then in both A and S c is interpreted by the same element. (4) If α ∈ P , then S = α. Given a pragmatic structure A, it may happen that there are no Anormal structures. It is possible to provide a system of necessary and sufficient conditions for the existence of Anormal structures (see Mikenberg et al. (1986), and da Costa (1974)). One condition of this system is as follows. For each partial relation Rk in A, we construct a set Mk of atomic sentences and negations of atomic sentences such that the former corresponds to ntuples that satisfy Rk , and the latter to ntuples that do not satisfy Rk (such sentences S correspond to ntuples in the “antiextension” of Rk ). Let M be the set k∈K Mk . Therefore, a sps A admits an Anormal structure only if the set M ∪ P is consistent. In what follows we shall always suppose that our sps satisfy the relevant conditions; in other words, given any sps A, the set of Anormal structures is not empty. DEFINITION 3. Let A be respectively a language and a sps in which L is interpreted. We say that a sentence α is pragmatically true in the sps A according to S if (1) A is a sps, (2) S is an Anormal structure, and (3) α is true in S (in conformity with the Tarskian definition of truth). That is, we say that α is pragmatically true in the sps A if there exists an Anormal S in which α is true (in the Tarskian sense). If α is not LOGI414.tex; 13/11/1998; 12:00; p.4 THE LOGIC OF PRAGMATIC TRUTH 607 pragmatically true in the sps A according to S (is not pragmatically true in the sps A), we say that α is pragmatically false in the sps A according to S (is pragmatically false in the sps A). 2. S TRICT PRAGMATIC VALIDITY Given a sps A, it is natural to consider its Anormal structures as the worlds of a Kripke structure for S5 with quantification, i.e., we have a universe and several structures, defined in such a universe, in which the language L can be interpreted, and where every world is accessible to every world (cf. Hughes and Cresswell (1968)). It is also natural to extend the language L of the sps A to a modal language, by the adjunction of the modal operator to its primitive symbols. The operator which in modal logic represents the notion of necessity, corresponds in the present situation to pragmatic validity (in a sps A). Analogously, the possibility symbol ♦, definable in terms of and negation, corresponds to pragmatic truth (in a sps A). Thus, we are led to extend the semantics of L in an obvious way, such that the symbols and ♦ will represent the concepts of pragmatic validity and of pragmatic truth, respectively. Moreover, since the universes of all “worlds” belonging to a sps are the same, it is reasonable that equality behaves, in the cases of pragmatic truth and of pragmatic validity, as necessary equality (cf. Hughes and Cresswell (1968), Chapter 11). Among the pragmatically valid formulas – that is, those formulas α such that α is a theorem of S5 with quantification and necessary equality −, there are the logically pragmatically true formulas – that is, those formulas α such that ♦α or, equivalently, ♦α is a theorem of the same system. From now on, in order to simplify the language, the former class of formulas will be called strictly pragmatically valid and the latter will be called pragmatically valid. The first class of formulas coincides with the set of theorems of S5 with quantification and necessary equality; the second, with Jaskowski’s logic associated with the same system. (We recall that, loosely speaking, given a modal system M, the Jaskowski’s logic associated to M is the set of all formulas a such that ♦α is a thesis of M; see Jaskowski (1969), Kotas and da Costa (1977) and da Costa (1975).)4 Therefore, the logical system which will be denoted by P V and which systematizes the notion of strict pragmatic validity has a language L∗ whose primitive symbols are those of a standard formalization of the firstorder predicate calculus with equality and individual constants, plus the symbol (for simplicity, function symbols are excluded). The defined symbols are introduced as usual, and the common conventions in the writing of for LOGI414.tex; 13/11/1998; 12:00; p.5 608 NEWTON C. A. DA COSTA ET AL. mulas, and in the formulation of postulates (axiom schemes and primitive rules of inference) etc. are employed without explicit mention. The postulates of P V are the following: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) If α is an instance of a (propositional) tautology, the α is an axiom. α, α → β/β (α → β) → (α → β) α → α ♦α → ♦α ∀xα(x) → α(t) α/α α → β(x)/α → ∀xβ(x) x=x x = y → (α(x) ↔ α(y)) In the postulates above, the symbols have clear meanings. In particular, in Axiom Scheme 6, t is either a variable free for x in α(x) or an individual constant. As we have already noticed, this system is essentially S5. We define the concept of deduction as in Henkin and Montague (1956). Their basic idea is essentially that one can only use the generalization rule – α/∀xα – in a step k of a deduction when a subsequence of the deduction up to k is a proof of α. This restriction to the generalization rule is exactly similar to the one adopted with regard to the necessitation rule, α/α. Then, the usual derived rules, such as the deduction theorem, remain valid.5 The semantics of P V can be easily developed: the basic (strict) semantical concepts of pragmatic truth, pragmatic falsehood, pragmatic validity, pragmatic invalidity, pragmatic semantic consequence etc. offer no difficulties in being formulated (and maintain the spirit of Definition 3). This claim notwithstanding, we reproduce here the definition of satisfiability. Let L∗ be a language and A a sps in which L∗ is interpreted. (L∗ is a modal firstorder language.) So, to each individual constant c of L∗ there is associated an element cA of the universe of A, and to each predicate symbol P of arity n of L∗ there is associated an nary partial relation PA , defined on the same universe. Moreover, with an Anormal structure S fixed, to each nary predicate symbol P of L∗ there is associated an nary total relation PS , and to each individual constant c of L∗ corresponds a member cS of the universe of S, such that cA = cS . An assignment (of L∗ in A) is a function f whose domain is the collection of individual variables of L∗ , which maps to each variable x an element f (x) of the universe of A. LOGI414.tex; 13/11/1998; 12:00; p.6 THE LOGIC OF PRAGMATIC TRUTH 609 If f is an assignment of L∗ in A, then f 0 is the function defined on the set of all variables and constants of L∗ , such that f 0 (t) is tA if t is a constant, and f 0 (t) is f (t) if t is a variable. Let A be a sps whose language is L∗ , and f be an assignment of L∗ in A; also let S be an Anormal structure. We define the relation hA, f, Si = α, or, in words, hA, f, Si satisfies the formula α of L∗ , by recursion on α: (1) If P is a predicate symbol of arity n and t1 , t2 , . . . , tn are n terms, then hA, f, Si = P (t1 , t2 , . . . , tn ) iff hf 0 (t1 ), f 0 (t2 ), . . . , f 0 (tn )i ∈ PS . (2) If ti and tj are two arbitrary terms, then hA, f, Si = ti = tj iff f 0 (ti ) = f 0 (tj ). (3) The common satisfaction conditions for the primitive connectives and the primitive quantifier. (4) If β is a formula, then hA, f, Si = β iff hA, f, Si = β, for all Anormal structure S. We have the following theorem, whose proof can be obtained by the methods of Hughes and Cresswell (1968) or of Gallin (1975): THEOREM 1. Let 0 be a set of formulas of L∗ and α be a formula of the same language. Then, 0 ` α if, and only if, 0 = α (where ` and = have clear meanings). That is, α is a syntactic consequence of 0 in P V iff α is a strict pragmatic semantic consequence of 0. The logic of strict pragmatic validity, which we have just sketched, can be extended to higherorder (modal) languages, for example by an adaptation of some ideas presented in Gallin (1975), Chapter 3. Moreover, one can also develop a metatheoretical study of this logic, for example, by adopting different modal systems as basic (S4, for instance), by distinguishing between frames and models etc. Instead of pursuing this line here, we will now consider a different system in order to study the logic of pragmatic validity. This system, as we will see, is constructed, as it were, in terms of P V , which was presented here mainly as an auxiliary construction. We will then show that the logic of pragmatic validity is paraconsistent. LOGI414.tex; 13/11/1998; 12:00; p.7 610 NEWTON C. A. DA COSTA ET AL. 3. P RAGMATIC VALIDITY The system P V constitutes a formalization of the notion of strict pragmatic validity. Now we present a system P T , whose language is the same as that of P V , which constitutes a formalization of the notion of pragmatic validity; the underlying intuition is that of constructing a system in which ` α means that ♦α is strictly pragmatically valid. Suppose that α is a formula of L∗ , the language of P V or of P T . We denote by ∀∀α any formula of the form ∀x1 ∀x2 . . . ∀xn α, where any free variable of α is one of the variables x1 , x2 , . . . , xn . (There may be variables in the sequence x1 , x2 , . . . , xn , which are not free in α.) When n = 0, we take ∀∀α to be α. Clearly, in order that α be pragmatically valid in the sense intended, we must have that ` α in P T if, and only if, ` ♦∀∀α in P V . So, P T is a kind of Jaskowski’s discussive logic associated with P V (see Jaskowski (1969), da Costa (1975), and Kotas and da Costa (1977)). By definition, then, α is a theorem of P T iff ♦α is a theorem of P V , and this definition directly furnishes a semantical interpretation for P T . P T can be axiomatized as follows: (10 ) If α is an instance of a (propositional tautology), then ∀∀α is an axiom. 0 (2 ) ∀∀α, ∀∀(α → β)/∀∀β (30 ) ∀∀((α → β) → (α → β)) (40 ) ∀∀(α → α) (50 ) ∀∀(♦α → ♦α) (60 ) ∀∀(∀xα(x) → α(t)) (70 ) ∀∀α/α (80 ) ∀∀α/∀∀α (90 ) ♦∀∀α/α (100 ) ∀∀(α → β(x))/∀∀(α → ∀xβ(x)) (110 ) Vacuous quantifications may be introduced or suppressed in any formula. 0 (12 ) ∀∀x(x = x) (130 ) ∀∀(x = y → (α(x) ↔ α(y))) In P T the definitions of proof and of (formal) theorem are the usual ones. LOGI414.tex; 13/11/1998; 12:00; p.8 THE LOGIC OF PRAGMATIC TRUTH 611 We proceed to show that postulates (10 )–(130 ) provide an axiomatization for P T . LEMMA 1. If α is a theorem of our proposed axiomatization for P T , then ♦∀∀α is a theorem of P V . Proof. By induction on the length of the proof of α in the proposed axiomatization for P T . Let α1 , α2 , . . . , αn , where αn is α, be a (formal) proof of α in the proposed axiomatization for P T . Then, αi , 1 6 i 6 n, is an axiom or is obtained by the application of one of the rules. If αi is an axiom, then it has the form ∀∀β, where β is an axiom of P V (observe that P V is S5 with quantification and necessary equality). Therefore, ∀∀β is a theorem of P V , and so ♦∀∀∀∀β, i.e., ♦∀∀αi is also a theorem of P V . Suppose that αi is a consequence of two preceding formulas by Rule 20 . Then αi is ∀∀β, obtained from the premises ∀∀γ and ∀∀(γ → β). By the induction hypothesis, ♦∀∀∀∀γ and ♦∀∀∀∀(γ → β) are provable in P V . Consequently, ∀∀γ and ∀∀(γ → β) are also provable in P V , and so is ∀∀β. But if ∀∀β is a theorem of P V , then, ♦∀∀∀∀β, 2 i.e., ♦∀∀αi , is also a theorem. The other rules are similarly treated. LEMMA 2. If α is a theorem of P V , then ∀∀α is a theorem of the proposed axiomatization for P T . Proof. By induction on the length of the proof of α in P V . Let α1 , α2 , . . . , αn , where αn is α, be a proof of α in P V . If αi , 1 6 i 6 n, is an axiom of P V , then ∀∀α is a theorem of the proposed axiomatization for P T , as is obvious. If αi is obtained by an application of modus ponens (Rule 2), from γ and γ → αi , we have, by the induction hypothesis, that ∀∀γ and ∀∀(γ → αi ) are provable in the proposed axiomatization. Then, by Rule 2, ∀∀αi is also provable. Rule 8 2 is treated analogously. THEOREM 2. Postulates (10 )–(130 ) characterize P T ; that is, we have: ` α in P T iff ` ♦∀∀α in P V . Proof. Let us suppose that α is a theorem of the proposed axiomatization for P T ; then, by Lemma 1, ♦∀∀α is a theorem of P V . LOGI414.tex; 13/11/1998; 12:00; p.9 612 NEWTON C. A. DA COSTA ET AL. Assume that ♦∀∀α is a theorem of P V . So, by Lemma 2, ∀∀♦∀∀α is a theorem of the proposed axiomatization for P T . Therefore, by Rule 70 , ♦∀∀α is a theorem of P T , and so by Rule 90 , α is also a theorem of P T . 2 DEFINITION 4. In P T we say that the formula α is a syntactic consequence of a set of formulas 0 (in symbols, 0 ` α) if there exist γ1 , γ2 , . . . , γn in 0, such that (♦γ1 ∧ ♦γ2 ∧ · · · ∧ ♦γn ) → ♦α is a theorem of P T (or, equivalently, ♦∀∀((♦γ1 ∧ ♦γ2 ∧ · · · ∧ ♦γn ) → ♦α) is a theorem of P V ). When n = 0, the first formula above reduces, by convention, to α (and ∅ ` α means, thus, that ` α). DEFINITION 5. A pragmatic theory is a set T of sentences (closed formulas of P T ), such that if γ1 , γ2 , . . . , γn are in T and {γ1 , γ2 , . . . , γn } ` α, then α is also in T . THEOREM 3. If T is a pragmatic theory and α is a (closed) theorem of P T , then α ∈ T . DEFINITION 6. Let E be the set of all sentences of P T and T be a pragmatic theory. T is called trivial (or overcomplete) if T = E; otherwise, T is called nontrivial. The theory T is called inconsistent if there is at least one sentence α such that α ∈ T and ¬α ∈ T , where ¬ is the symbol of negation of P T ; otherwise, T is called consistent.6 THEOREM 4. There exist pragmatic theories which are inconsistent but nontrivial. Proof. Let c and M be respectively any individual constant and a monadic predicate symbol of P T . The theory whose (nonlogical) axioms are M(c) and ¬M(c) is inconsistent. But it is nontrivial, because the corresponding theory of P V , whose (nonlogical) axioms are ♦M(c) and ♦¬M(c), is consistent. In effect, it is easy to construct a Kripke model for P V in which both ♦M(c) and ♦¬M(c) are true. However, in no Kripke model for P V the formula ♦(M(c) ∧ ¬M(c)) is true. (See also Theorem 9 of da Costa 2 (1975).) DEFINITION 7. If α and β are formulas, we put α →d β stands for ♦α → β; α ∧d β stands for ♦α ∧ β. LOGI414.tex; 13/11/1998; 12:00; p.10 THE LOGIC OF PRAGMATIC TRUTH 613 The connectives →d and ∧d are called discussive implication and discussive conjunction, respectively. THEOREM 5. In P T →d , ∧d and ∨ satisfy all the valid schemes and rules of classical positive logic. Proof. If we consider a valid primitive scheme (or rule) of classical positive logic, and replace in it implication by discussive implication and conjunction by discussive conjunction, we obtain a valid scheme (or rule) of P T , 2 as is easily seen. THEOREM 6. If T is a pragmatic theory, then α ∈ T iff there exist γ1 , γ2 , . . . , γn in T such that (γ1 ∧d γ2 ∧d · · · ∧d γn ) →d ♦α is a theorem of P T . Da Costa (1975) contains a different formulation of the discussive logic associated with P V , which we will dub here P T ∗ . P T ∗ is defined as follows: α is a theorem of P T ∗ iff ♦α is a theorem of P V . It is clear that our definition of the discussive logic associated with P V is better than P T ∗ from the point of view of pragmatic truth. We have: THEOREM 7. If α is a theorem of P T , then α is also a theorem of P T ∗ . Proof. In effect, if ` α in P T , then, by definition, ` ♦∀∀α in P V ; but in P V we have: ` ♦∀∀α → ∀∀♦α. Hence, in P V , ` ∀∀♦α, and ` ♦α, since in the last system the scheme 2 ∀∀α → α is valid. THEOREM 8. There are theorems of P T ∗ which are not theorems of P T . Proof. The proof consists in the presentation of a formula α such that ` α in P T ∗ , i.e., ` ♦α in P V , but it is not the case that ` α in P T , i.e. it is not the case that ` ♦∀∀α in P V . Let α be the formula ♦M(x) → M(x), where M is a monadic predicate symbol. We have: (1) Since ♦(♦p → p), where p is a propositional variable, is a theorem of S5, we deduce that in P V ` ♦(♦M(x) → M(x)). LOGI414.tex; 13/11/1998; 12:00; p.11 614 NEWTON C. A. DA COSTA ET AL. Therefore, ♦M(x) → M(x) is a theorem of P T ∗ . (2) It is not the case that ` ♦∀∀x(♦M(x) → M(x)) in P V . The verification of this assertion consists of obtaining a Kripke structure for P V in which the following formula becomes true: ¬♦∀x(♦M(x) → M(x)) that is ∃x(♦M(x) ∧ ¬M(x)). We consider a Kripke model hW, D, V i (see Hughes and Cresswell (1968)), where W = {w1 , w2 }, D = {a, b}, V (M, w1 ) = {a} and V (M, 2 w2 ) = {b}. This structure satisfies the preceding formula. We observe that in the applications of the theory developed above, it is sometimes convenient to employ an alternative definition of syntactic consequence. For instance, in certain applications of our ideas in the domain of the foundations of physics, instead of Definition 4, it is more appropriate to adopt the following one: DEFINITION 40 . In P T , the sentence α is said to be a proper syntactic consequence of a set of sentences 0 if there is γ1 , γ2 , . . . , γn in 0, such that ♦(γ1 ∧ γ2 ∧ · · · ∧ γn ) and ((γ1 ∧ γ2 ∧ · · · ∧ γn ) → γ ) are theorems of P T (or of P T and some extra axioms). The corresponding logic developments and applications will be treated in forthcoming works. 4. P HILOSOPHICAL CONSIDERATIONS The philosophical significance of the above formal account can be drawn out through a consideration of inconsistency in our belief systems. If one focuses on Theorem 4, it can be seen that a pragmatic theory can contain contradictory theorems without reducing to triviality. This means that P T belongs to the class of paraconsistent logics, since a logical system is paraconsistent if it can be employed as the underlying logic of inconsistent but nontrivial deductive systems or theories (for a useful outline of the nature of paraconsistent logics, see Arruda (1980)). In the case of pragmatic truth, this is not an unreasonable situation: contradictory propositions may, of course, both be pragmatically true. Thus pragmatic truth can be used to provide the epistemic framework for characterizing inconsistent belief systems. LOGI414.tex; 13/11/1998; 12:00; p.12 THE LOGIC OF PRAGMATIC TRUTH 615 The occurrence of inconsistent theories in science, for example, is now a widely recognized phenomenon; examples range from Bohr’s theory of the atom and the “old” quantum theory of blackbody radiation to the infinitesimal calculus and Stokes’ analysis of the motion of a pendulum. The problem, of course, is how to accommodate this aspect of scientific practice given that within the framework of classical logic an inconsistent result is disastrous: the set of consequences of an inconsistent theory will explode into triviality and the theory is rendered useless. Another way of expressing this descent into logical anarchy is to say that under classical logic the closure of any inconsistent set of sentences includes every sentence. It is this which lies behind Popper’s famous declaration that the acceptance of inconsistency “[. . .] would mean the complete breakdown of science” (Popper (1940)). Various approaches for accommodating inconsistent theories can be broadly delineated. Thus one can distinguish “logic driven control” of this apparent logical anarchy from “content driven control”, where the former involves the adoption of some underlying nonclassical logic and the latter focuses on the specific content of the theory in question (Norton (1993)). A well known example of the former approach is the work of Priest, who cites the existence of inconsistency in science as the pragmatic “fulcrum” of his approach, levering the reader into acceptance of the Hegelian view that there exist true contradictions, where “true” is here understood in the correspondence sense (Priest (1987)). The classical collapse of an inconsistent theory into triviality is then prevented through the adoption of a form of paraconsistent logic in which certain contradictions are tolerated. That such an approach fails to do justice to the doxastic attitude of the scientists developing such theories is clear: noone believed that Bohr’s theory was true, not even Bohr himself (see da Costa and French (forthcoming)). According to he alternative “content driven” approach, the attitude of scientists to inconsistency is based simply on “[. . .] a reflection of the specific content of the physical theory at hand” (Norton (1993), p. 417). Thus, it is argued, from an inconsistent theory we can construct a consistent subtheory yielding the relevant empirically supported laws. However, that this is possible with hindsight is irrelevant to the question of what attitude should be adopted towards inconsistent theories before such consistent reconstructions have been identified (see Brown (1990), p. 292; Brown (1993); Smith (1988a), and Smith (1988b)). An alternative strand within this approach is to regard inconsistent theories as “prototheories” and shift away from a concern with truth to an epistemic attitude of “entertainment” (Smith (1988a)). Inconsistency can then be accommodated by abandoning LOGI414.tex; 13/11/1998; 12:00; p.13 616 NEWTON C. A. DA COSTA ET AL. closure for such “prototheories”, in the sense that the latter can be broken up into (self) consistent subsets from each of which implications can be derived in classical fashion. Abandoning closure in order to accommodate inconsistency is a well known move, of course. Thus Kyburg has remarked that “it is not the strict inconsistency of the rational corpus that leads to trouble – it is the imposition of deductive closure” (Kyburg (1987), p. 147). Smith, in particular, follows Harman in claiming that accepting deductive closure should not be confused with inference (Smith (1988a)). Harman in turn dismisses the “Logical Closure Principle” on the grounds that closure is simply an unrealistic requirement (see Harman (1986), pp. 12–14). Together with the acknowledged defeasibility of the claim that inconsistency is to be avoided, such arguments are adduced to support the more general line that “[. . .] there is no clearly significant way in which logic is specially relevant to reasoning” (ibid., p. 20). What began as a maneuver to accommodate inconsistency while staying outside a paraconsistent framework has terminated in a position of such extremity that one can only wonder if the supposed cure is worse than the disease. Furthermore, without the imposition of closure, there seems nothing to prevent the division into consistent subsets from becoming an unsystematic affair, with “anything goes” being the order of the day with regard to the application of claims from these subsets. Finally, talk of “entertaining” Bohr’s model as “prototheory” is inappropriate, given the level of commitment maintained in its exploitation and further development. Such commitment is the hallmark of full blooded acceptance and what is required is some way of doing justice both to this attitude and the view that inconsistent theories should not be regarded as true in the correspondence sense. There is a great deal more to be said here, of course (see da Costa and French (forthcoming)), but our claim is that “doing justice” to such doxastic attitudes involves the elaboration of an alternative account of belief according to which “belief that p” is not to be understood as “belief that p is true”, in the correspondence sense. Such an account has been developed in da Costa and French (1989); (1993a); (1993b); and French (1991), where the view is defended that, when it comes to representational structures such as scientific theories, “belief that p” is to be understood as “belief that p is pragmatically or partially true”. This allows for the accommodation of inconsistency by acknowledging that it is not a permanent feature of reality to which theories must correspond, but is rather a temporary aspect of such theories which may nevertheless be extremely fruitful in a heuristic sense (see da Costa and French (1993a) and da Costa, LOGI414.tex; 13/11/1998; 12:00; p.14 THE LOGIC OF PRAGMATIC TRUTH 617 Bueno and French (forthcoming)). On this account it is not the “logic of science”, in the sense of the underlying logic of deduction and inference, which is paraconsistent, but rather the appropriate “logic of truth”. Relatedly, the logic of pragmatic truth as delineated above may serve as a “logic of scientific acceptance” (see da Costa and French (1993a)). The nature of acceptance is relatively little discussed within the philosophy of science. Those accounts that do consider it tend to divide between two extremes: those that identify acceptance and belief and those that separate the two entirely. The former typically regard belief in terms of the correspondence view of truth, whereas the latter fall prey to the accusation of conventionalism in theory choice (for further discussion, see da Costa and French (1993a)). An alternative “via media” is to retain the connection between belief and acceptance whilst rejecting truthascorrespondence. In this view, to accept a theory is to be committed, not to believing it to be true per se, but to holding it as if it were true, for the purposes of further elaboration, development and investigation. Thus acceptance involves belief that the theory is partially or pragmatically true only and this, we believe, corresponds to the fallibilistic attitude of scientists themselves.7 Linking acceptance and pragmatic truth in this way restores a formal similarity between “truth”, taken generally, and acceptance with regard to deductive closure. It has been argued, for example, that acceptance differs from truth in that whereas the latter is deductively closed, in the sense that what one deduces from a set of truths is also true, the former generally is not (see Ullian (1990)). This is correct if closure is understood only in classical terms. However, what the above formal analysis shows is that acceptance, understood within the framework of pragmatic truth, may be regarded as closed under the Jaskowski’s discussive system (da Costa and French (1993a)). To put it more precisely: although there is no closure under classical conjunction and material implication, one can define discussive forms of implication and conjunction as above, with respect to which acceptance can indeed be regarded as closed. This is a result of both general and particular significance. Our contention is that inconsistency can be accommodated within an appropriate framework under which the set of propositions which we accept is closed under implication. The appropriate framework is precisely that which we have presented in this paper and the form of implication is, of course, discussive. Shifting perspective again from the specific to the more general, it is the failure to consider such nonclassical systems which undercuts the claim that “logic” is not specially relevant to reasoning. Within the framework of pragmatic truth we can accommodate inconsistency while LOGI414.tex; 13/11/1998; 12:00; p.15 618 NEWTON C. A. DA COSTA ET AL. still retaining a sense of deductive closure. In this manner the relevance of logic to reasoning – especially scientific reasoning – is restored. N OTES 1 The authors wish to acknowledge the helpful comments of two anonymous referees for this journal. 2 Interesting enough, Colin Howson, at a recent meeting of the British Society for the Philosophy of Science, in which one of us (S. French) was presenting a piece (French (1997)), asked for the connections between the logic of pragmatic truth and Kripkean semantics. Though this paper was already written when this meeting took place, it certainly supplies an answer to Howson’s question. 3 More formally, an nplace partial relation R can be viewed as a triple hR , R , R i, 1 2 3 where R1 , R2 , and R3 are mutually disjoint sets, with R1 ∪ R2 ∪ R3 = D n , and such that R1 is the set of ntuples that belong to R; R2 the set of ntuples that do not belong to R; and finally R3 of those ntuples for which it is not defined whether they belong or not to R. (Note that when R3 is empty, R is a normal nplace relation that can be identified with R1 .) 4 Related works in Jaskowski’s logic are: D’Ottaviano and da Costa (1970), D’Ottaviano (1985), D’Ottaviano and Epstein (1988), Avron (1991), and Epstein (1995), Chapter IX. 5 It should be noticed that in our system the Barcan formula is a theorem. This comes, in particular, from the fact that Anormal structures have the same universe (see Definition 1). 6 We assume, in this context, that there are essentially two negations. In paraconsistent situations, we have to use the weak negation ¬; in classical situations, we may use the strong negation ¬♦α which behaves in the same way as the classical negation of α. The gist of our logic is to handle both negations according to our needs. 7 The formalism articulated here can also be used in defense of an empiricist interpretation of science, since it provides tools for an agnostic view about theoretical terms (as is the case of van Fraassen’s constructive empiricism; see van Fraassen (1980) and (1989)). Consider, for instance, the structures discussed in Section 1. Roughly speaking, we use those structures with domain A1 to accommodate observable features of physical systems, and those with domain A2 to describe nonobservable, theoretical properties introduced in the study of such systems. The set of accepted sentences P puts constraints on the extensions of the pragmatic structures under consideration. In this context, a sentence α is pragmatically true if it is true (in the Tarskian sense) under one possible assignment of values to the theoretical terms which agrees with the accepted information in P and with the observable phenomena in A1 . This assignment is done, of course, in a convenient Anormal structure. Further discussion about this issue can be found in da Costa and French (1990), Bueno (1996) and Bueno (1997). R EFERENCES Arruda, A. I. (1980): A Survey of Paraconsistent Logic, in: A. J. Arruda, N. C. A. da Costa and R. Chuaqui (eds.), Mathematical Logic in Latin America, NorthHolland, Amsterdam, pp. 1–41. LOGI414.tex; 13/11/1998; 12:00; p.16 THE LOGIC OF PRAGMATIC TRUTH 619 Avron, A. (1991): Natural ThreeValued Logics – Characterization and Proof Theory, The Journal of Symbolic Logic 56: 276–294. Brown, B. (1990): How to be Realistic about Inconsistency in Science, Studies in History and Philosophy of Science 21: 281–294. Brown, B. (1993): Old Quantum Theory: A Paraconsistent Approach, PSA 1992, vol. 2, East Lansing, Philosophy of Science Association, pp. 397–411. Bueno, O. (1996): What is Structural Empiricism? Scientific Change in an Empiricist Setting, paper read at the 1996 Annual Congress of the British Society for the Philosophy of Science, Sheffield, 11–13 September, forthcoming. Bueno, O. (1997): Empirical Adequacy: A Partial Structures Approach, forthcoming in: Studies in History and Philosophy of Science. da Costa, N. C. A. (1974): αmodels and the systems T and T∗ , Notre Dame Journal of Formal Logic 14: 443–454. da Costa, N. C. A. (1975): Remarks on Jaskowski’s Discussive Logic, Reports in Mathematical Logic 4: 7–16. da Costa, N. C. A. (1986): Pragmatic Probability, Erkenntnis 25: 141–162. da Costa, N. C. A., Bueno, O. and French, S. (forthcoming): Paraconsistency and Partiality, in preparation. da Costa, N. C. A. and French, S. (1989): Pragmatic Truth and the Logic of Induction, The British Journal for the Philosophy of Science 40: 333–356. da Costa, N. C. A. and French, S. (1990): The ModelTheoretic Approach in the Philosophy of Science, Philosophy of Science 57: 248–265. da Costa, N. C. A. and French, S. (1993a): Towards an Acceptable Theory of Acceptance: Partial Structures, Inconsistency and Correspondence, in: S. French and H. Kamminga (eds.), Correspondence, Invariance and Heuristics, Kluwer Academic Publishers, Dordrecht, pp. 137–158. da Costa, N. C. A. and French, S. (1993b): A Model Theoretic Approach to “Natural Reasoning”, International Studies in the Philosophy of Science 7: 177–190. da Costa, N. C. A. and French, S. (forthcoming): Partial Truth and Partial Structures: A Unitary Account of Models in Scientific and Natural Reasoning, unpublished book, University of São Paulo and University of Leeds, in preparation. D’Ottaviano, I. M. L. (1985): The Model Extension Theorems for J3Theory, in C. A. di Prisco (ed.), Methods in Mathematical Logic, Springer, New York, pp. 157–173. D’Ottaviano, I. M. L. and da Costa, N. C. A. (1970): Sur un problème de Jaskowski, Comptes Rendus de l’Académie des Sciences de Paris 270A: 1349–1353. D’Ottaviano, I. M. L. and Epstein, R. (1988): A Manyvalued Paraconsistent Logic, Reports on Mathematical Logic 22: 89–103. Epstein, R. (1995): Propositional Logics, 2nd edition, Oxford University Press, Oxford. French, S. (1991): Rationality, Consistency and Truth, The Journal of NonClassical Logic 7: 51–71. French, S. (1997): Partiality, Pursuit and Practice, in: Dalla Chiara et al. (eds.), Structures and Norms in Science, Kluwer Academic Publishers, Dordrecht, pp. 35–52. Gallin, D. (1975): Intensional and HigherOrder Logic, NorthHolland, Amsterdam. Harman, G. (1986): Change in View, MIT Press, Cambridge, Mass. Henkin, L. and Montague, R. (1956): On the Definitions of Formal Deduction, The Journal of Symbolic Logic 21: 129–136. Hughes, G. H. and Cresswell, M. J. (1968): An Introduction to Modal Logic, Methuen. Jaskowski, S. (1969): Propositional Calculi for Contradictory Deductive Systems, Studia Logica 24: 143–157. LOGI414.tex; 13/11/1998; 12:00; p.17 620 NEWTON C. A. DA COSTA ET AL. Kotas, J. and da Costa, N. C. A. (1977): On Some Modal Logical Systems Defined in: Connection with Jaskowski’s Problem, in A. I. Arruda, N. C. A. da Costa, and R. Chuaqui (eds.), NonClassical Logic, Model Theory and Computability, NorthHolland, Amsterdam, pp. 57–73. Kyburg, H. (1987): The Hobgoblin, The Monist 70: 141–151. Mikenberg, I., da Costa, N. C. A., and Chuaqui, R. (1986): Pragmatic Truth and Approximation to Truth, The Journal of Symbolic Logic 51: 201–221. Norton, J. (1993): A Paradox in Newtonian Gravitation Theory, PSA 1992, vol. 2, East Lansing, Philosophy of Science Association, pp. 412–420. Peirce, C. S. (1965): Philosophical Writings of Peirce, selected and edited by J. Buchler, Dover, New York. Popper, K. R. (1940): What is Dialectic?, Mind 49: 403–426. Priest, G. (1987): In Contradiction: A Study of the Transconsistent, Martinus Nijhoff, Dordrecht. Smith, J. (1988a): Scientific Reasoning or Damage Control: Alternative Proposals for Reasoning with Inconsistent Representations of the World, PSA 1988, vol. 1, East Lansing, Philosophy of Science Association, pp. 241–248. Smith, J. (1988b): Inconsistency and Scientific Reasoning, Studies in History and Philosophy of Science 19: 429–445. Ullian, J. S. (1990): Learning and Meaning, in R. B. Barrett and R. F. Gibson (eds.), Perspectives on Quine, Oxford, Blackwell, pp. 336–346. Vaihinger, H. (1952): Philosophy of “As If” A System of Theoretical, Practical and Religious Fictions of Mankind, Routledge & Kegan Paul, London. van Fraassen, B. C. (1980): The Scientific Image, Clarendon Press, Oxford. van Fraassen, B. C. (1989): Laws and Symmetry, Clarendon Press, Oxford. NEWTON C. A. DA COSTA Department of Philosophy, University of São Paulo, São PauloSP, 05508900, Brazil (Email: ncacosta@usp.br) OTÁVIO BUENO and STEVEN FRENCH Division of History and Philosophy of Science, Department of Philosophy, University of Leeds, Leeds, LS2 9JT, UK (Email: phloab@leeds.ac.uk) LOGI414.tex; 13/11/1998; 12:00; p.18